A NNALES SCIENTIFIQUES DE L ’É.N.S.
M OTOO U CHIDA
Microlocal analysis of diffraction by a corner
Annales scientifiques de l’É.N.S. 4e série, tome 25, no1 (1992), p. 47-75
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4s serie, t. 25, 1992, p. 47 a 75.
MICROLOCAL ANALYSIS OF DIFFRACTION BY A CORNER
BY MOTOO UCHIDA
Introduction
The purpose of the present paper is to study analytic singularities of solutions to mixed type boundary value problems in the exterior domain of a corner. In particular, we prove that the cone of diffracted singularities is produced by an incident ray which hits the corner. This phenomenon has been observed without proof by J.-B. Keller in his geometrical theory of diffraction [II], where he conjectured that his diffracted ray method does yield the leading terms in the asymptotic expansions of solutions of diffraction problems. In this paper, we shall give a proof to his observation in the analytic category.
Let M be a real analytic manifold. Let Q^, Q^ ^e op^11 subsets of M given by (pi >0, cp2>0 respectively for real-valued OMunctions (pi, cp^ with ^cpi Aafcp^O. Let
Q = O i U Q 2 -
To every hyperfunction u defined on'Q is associated the closed conic subset SSo(^) of T^X, with X being a complex neighborhood of M; this set is called the boundary analytic wavefront set of u (which was introduced by P. Schapira [19], [20], [21]). The singular spectram (or the analytic wavefront set) SS (u) of u over Q is a closed conic subset of Q x ^ T ^ X and the equality SS^(u) C}n~1 (Q)=SS(^) holds, where T I : T ^ X - ^ M . [C/. 2 . 4 for the definition of the set SS^ (u).]
The main purpose of this paper is to prove the following "propagation" and "condensa- tion" results of analytic singularities at a corner. Let Q = Q i U ^ 2 ^e as above; set K = M\Q, No = { x e M (pi (x) = (p^ (x) = 0}. Let P = P (x, D) be a differential operator with analytic coefficients on M with principal symbol /=/(z, Q. Let peT^X with 7i(/?)eNo. Assume
(0.1) each boundary { ( p i = 0 } , { ( p 2= =0 } o f Q ^ , 0^ ls noncharacteristic for P;
(0.2) Im/|T^X=0;
(0.3) df/\ co (p) 7^0, with co being the fundamental 1-form on T* X.
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Let b± (q) denote the positive (resp. negative) bicharacteristic curve of P on T^X issued from q e T^ X with / (q) = 0. Assume in addition
(0.4) b± (p) is transversal to K and n (^± (7?)) c= Q
[cf. (6.5) for the precise meaning]. Let YQ = { z eX \ (pi (z) = (p^ (z) == 0}, po : Yo x x T* X
^ T* YQ the natural projection. Define a real curve passing through p:
C ^ { ^ e N o X ^ X | / ( ^ ) = 0 , po^-PoW Then we have:
THEOREM 0 . 1 . — Z ^ r u be a hyper function solution on Q. to the differential equation Pu=0. If b~(p)^SS(u) and if b~ {q) U SS(u)=0 for every qeCp\{p] close to p,
then p e SS^ (u) and b + (p) c SS (u).
THEOREM 0 . 2 . — L e t u be as in Theorem 0.1. If p e SS^ (u)\SS (u)C\b~ (p), then there is a neighborhood Cp(o) ofp in Cp such that Cp (s) c SS^ (u) and b+ (q) c: SS (u) for allqeC^).
Theorem 0.1 asserts that an isolated singularity ofu propagates beyond the corner along the bicharacteristic curve. On the other hand, Theorem 0.2 proves the appearance of diffracted rays at the corner. These theorems are proved in Section 6 (6.2 and 6.3).
In Section 7, we shall apply Theorems 0.1 and 0.2 to the Dirichlet problem in the region Q for a second order differential operator P of real principal type. Preparing a lemma of the reflection of singularities at the corner (cf. 7.2.2), we prove (cf. 7 . 3 . 2 for the precise statement):
THEOREM 0.3. — Let u be a solution to the mixed type Dirichlet boundary value problem for P in Q. If u has a single incoming singularity "in general position" at the corner,
then u has the outgoing singularities forming the cone of diffracted rays.
The argument used there clarifies the microlocal geometrical aspect of diffraction problems, and this gives a proof to Keller's geometrical theory [11] of diffraction by a corner from the standpoint of microlocal analysis. Cf. also the work of Cheeger and Taylor [I], Rouleux [14], Varrenne [27] for diffraction of a simple progressing wave on a Riemannian manifold by a conical singularity {cf. 7.3.4).
Our method is based on the theory of ^A|X of R Schapira ([19], [20], [21]), which has been propounded as a framework for microlocal study of boundary value problems in a general domain (possibly with non smooth boundary). Summarizing some generalities of sheaves in Section 1, we make a short review on this general theory in Section 2. Cf.
also, e.g., the work of Kataoka [7], [8], [10], Schapira [17] to [20], Sjostrand [23] for microlocal analysis in the analytic category of boundary value problems in domains with smooth boundary. Section 3 is a supplement to the general theory of ^A|X-
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In Section 4, we prove the injectivity of a certain homomorphism induced from inclusion of open subsets; this will be used in the proofs of Theorems 0.1 and 0.2. This section actually constitutes the central part of the proof of diffraction.
Section 5 is a preparatory section to the succeding sections. In Section 6, we prove Theorems 0 . 1 and 0.2. In Section 7, we state Theorem 0.3 in precise form and prove it by applying Theorems 0.1 and 0.2.
The main results of this paper have been announced in [26].
The author would like to thank P. Schapira and K. Kataoka. Not only is this work based on the theory of ^x °f P- Schapira for the framework, but also the proof of Theorem 3.1 has been completed on his suggestion. The author also benefitted much from discussions with K. Kataoka at a preliminary stage of this work. The author must thank A. Kaneko for pointing out Keller's paper. Finally, the author would like to express his sincere gratitude to H. Komatsu for his constant encouragement.
1. Generalities of microlocal study of sheaves
Let X be a C°°-manifold, n: T* X -> X its cotangent bundle. D (X) denotes the derived category of the category of complexes of sheaves of C-vector spaces on X, and D^ (X) denotes the full subcategory consisting of complexes with bounded cohomologies. In this section we recall some basic notions of microlocal study of sheaves; refer to [6] for the details.
1.1. For two subsets S^, S^ of X, the normal cone C^(Si; S^) of S^ along S^ at x e X is the subset of T^X defined by
C^(Si; S 2 ) = { ^ e T ^ X | t h e r e are sequences { x ^ } c = S i , { ^ n ^ S ^ ,
and { ^ } c [ R + such that x^-^x, y^->x, a^(x^-y^)->v}.
For a subset S of X and x e X, we set
N,(S)=T,X\C,(X\S;S),
N ; ( S ) = { O e T ; ? X | < e , z ; > ^ 0 for all z;eN,(S)}.
We denote by N*(S) the union of N^(S)(xeX), which is a closed convex conic subset of T* X and called the conormal cone to S in X.
1.2. The constant sheaf on X is denoted by Cx. For a locally closed subset A of X, we set CA = i\ i~1 Cx with i: A c^ X.
1.3. For an object F of Db(X), SS(F) denotes the microsupport of F due to Kashi- wara-Schapira [6], which is by definition, roughly speaking, the set of codirections of X in which codirections F does not propagate. The microsupport SS (F) is a closed conic involutive subset of T* X.
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Let K be a closed subset of X. K is said to be C^convex ( l ^ a ^ o o ) at x if K is convex for a choice of local C^-coordinates of X in a neighborhood of x.
PROPOSITION 1.1. — Let K be a C^-convex closed subset ofX, let T^X=SS(CK). Let (x; Q e T* X. Then (x; ^) e T^ X ;/ and only if there is a C^-function g on X with g (x) = 0, dg(x}=(x, Q and K < = { ^ ^ 0 } in a neighborhood of x. In particular, each fibre o/T^X is convex.
Considering the above proposition, T^X is called the generalized conormal set of K.
PROPOSITION 1.2. — Let U be an open subset ofX; let TgX=SS(Cu). Then we have T g X c U X x N ^ U )0,
where (•)" denotes the antipodal map on T*X. IfX\\J is C1-convex and U = I n t U , equality holds.
Let M be a closed submanifold of X, (p : M -> X the embedding. Let p and m be the natural maps associated with (p:
T * M ^ — M x ^ T * X ^ ^ T * X .
PROPOSITION 1.3. — Let G be an object o/D^M). Then SS((p„G)=T^p-l(SS(G)).
In particular, SS((p^G) is T^X-invariant.
COROLLARY 1.4. — Let Q be an open subset ofM; let T^X= SS (C^). Then T^XczTup-^Qx^N^Qn.
1.4. Let M be a closed submanifold of X. Sato's microlocalization along M is denoted by j^O^) -^(T^X). Refer to Sato-Kawai-Kashiwara [15], chap. 1, and Kashiwara-Schapira [6], chap. 2, for its construction and fundamental properties.
There is also the bifunctor (cf. [6])
\
Uhom: D^ (X)° x D^ (X) -^ D^ (T* X), which possesses the following properties:
(1.1) Hhom(CM,F)^MF), (1.2) R 7^ jLihom (G, F) ^ R ^om (G, F), (1.3) supp (nhom (G, F)) c= SS (G) U SS (F).
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In particular,
(1.4) supp (ahom (C^, F)) c SS (F) U T^ X for Q c M c: X of Corollary 1.4.
For /?eT*X, we denote by D^X;^) the localization of D^X) by the null system {GeOb(Db(X))|7^SS(G)}. It follows from (1.3) that if G^ and G^ are isomorphic in D^X;/?), then ahom(Gi, F) and ahom (G2, F) are isomorphic in a neighborhood of p.
2. Review on the theory of C\|x
Let M be a real analytic manifold of dimension n, X a complex neighborhood of M, n: T* X -> X the cotangent bundle of X. We use the following notations for sheaves:
(9^\ ^e sheaf of holomorphic functions on X;
ja^M: the sheaf of real analytic functions on M (= 0^ |^);
^: ^e sheaf of Sato's hyperfunctions on M;
^M: the sheaf of Sato's microfunctions on Tj^X;
Q)^. the sheaf of rings of differential operators of finite order on X;
^x^ the sheaf of rings of microdifferential operators of finite order on T* X.
Refer to Sato-Kawai-Kashiwara [15] for the definitions and fundamental properties of
^ ^ ^x. <^x- <y also [4], [22]. We denote by D^TC"1 ^x) ^ derived category of the category of complexes of n~1 ^^-modules with bounded cohomologies.
Let A be a locally closed subset of M. Following Schapira ([19], [20]), we define an object ^|x of D& (7C ~1 ^x) ^
(2.1) <^|x=^om(CA, ^x)®orM,xM,
with or^ix being the relative orientation sheaf of M in X. In particular, for A = M ,
^M|X ls nothing but the sheaf ^ of Sato's microfunctions.
In [19] and [20], by using the functorial definition (2.1), a general framework is set up for microlocal study of boundary value problems. In this section we quickly recall the notations and results in the theory of ^ix- c/ t19]. t20]. E2!] ^ th^ details of this section; cf. [25] for Sections 2.8 and 2.9.
2.0. Let A be a locally closed subset of M. Since the flabby dimension of ^x ls ^^
H^Ap^OforoO.
2.1. Let K be a closed C^-convex (i. e. convex for a choice of local coordinates) subset of a real analytic manifold M.
PROPOSITION 2.1. — The complex of sheaves ^K|X is concentrated in degree 0;
WK|X)=O (^0);
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therefore ^iqx ln an ^ ^-module, and the natural ^\-homomorphism ^ K i x I r M X ^ ^ M ztsl
infective.
Remark. — The d^-module ^K|X was ^lrst introduced in a different way by Kataoka ([7], [8], [9]) in the case where K is a closed half-space of M with C^-boundary. The functorial definition (2.1) is due to Schapira ([19], [20]).
Let Q=M\K. Then there exists a distinguished triangle (2.2) ^KIX^M^^IX—^.
Hence ^x 1s quasi-isomorphic to a complex of ^^-modu\es, and we have:
COROLLARY 2.2. — H° (^x) is supported in T^ X, and H1 (^,x) ITM x = ° f^ ^°- 2.2. Let Q, 0.' be two open subsets of M with Q ^ Q'. There is a canonical morphism ("microlocal restriction")
(2.3) ^Qjx -> ^n'jx-
2.3. Let Q be an open subset of M. Assume that N^(Q)^T^M for all xeO. This implies that, for a choice of local coordinates, there is an open convex cone y such that Q + y c Q locally. We say that Q has the cone property if Q. satisfies this condition.
Let N be a closed submanifold of M of codimension d. If N c: Q, then we have the boundary value morphism due to Schapira:
(2.4) bv: ^^N|x®orN|MM, where or^M denotes the relative orientation sheaf of N in M.
Remark. — There is a topological boundary value morphism CN®CON|M -> ^ where
^NiM^^NiMt""^]- Schapira [19] constructed morphism (2.4) by applying the functor ahom (•, (9^) to the topological boundary value morphism.
2.4. Let Q be an open subset of M. We have the spectral map oc: K^r^^-.H0^).
Let ^er(Q, ^^). We set SS^(^)=supp(oc(^)); this set is a closed conic subset o f T ^ X and called the boundary analytic wavefront set of u.
2.5. Let M be a coherent Q)x-module (i. e. a system of differential equations); Char (^) denotes the characteristic variety of^. Let Q be an open subset of M and let T^X denote the microsupport of CQ. We have the spectral map
a: n ~1 F^ J^om^ (^, ^) ^ H° R ^om^ (^, ^x).
Let Mer(Q, ^om^^M, ^yS). We set SS^f (u) = supp (a (u)), which is a closed conic subset of T^ X n Char (^).
4eSERIE - TOME 25 - 1992 - N° 1
Remark. — Let ^=^x/^xP- ^ M\Q is C^-convex in M, we have the equality SS^f (u) 0 T^ X = SS^ (u) for any hyperfunction solution u to P on Q. This follows from Corollary 2.2.
2.6. Let ^ be a coherent ^x"1110^11!6- Let Q, 0' be open subsets of M with Q=3Q\ Then we have the map [cf. (2.3)].
(2.5) P: H° R ^om^ (^, ^) -^ H° R ^m^ (^, ^,x).
Let ^ be a section of H° R ^fom^ ( J / , ^jx)- we set ss^ (^)= ^PP (P <^))-
2.7. Let N be a closed C^-submanifold of M, Y the complexiflcation of N in X. Let p and TO be the natural maps associated with Y c^ X:
T* Y^—Y x ^ T* X—^T* X.
Let Ji be a coherent ^x"1110^^ and suppose that Y is non characteristic for ^. The tangential system of M on Y is denoted by ^y-
PROPOSITION 2.3. — For a locally closed subset A ^/N, there is a natural isomorphism (2.6) p^^R^m^^f, ^A\x)®or^[d]^R^om^(^ ^ ^\
with J==codimN, where ^[Y denotes the complex of Q) ^-modules defined in the same way as is ^|x m (^-1)'
Let Q be an open subset of M having the cone property. If N <= Q, then, using (2.4) and (2.6) with A = N, we obtain the boundary value morphim
(2.7) bv: p^~1 R ^om^ (^, ^,x) ^ R ^fom^ (^y, ^i^)-
PROPOSITION 2 . 4 (Schapira [19]). — Assume that the boundary ofQ. is a C^-hyper surf ace and Q. lies on one side of it. Let N==3Q. If N is non characteristic for Ji, then the homomorphism [cf. (2.4)]
bv: H° R ^om^ (^, ^|x) ^ H1 R F^xnTNx R ^om^ (^ ^N|x)®orN)M
^ injective on T^X 0 T^X; therefore (2.7) ^ ^fao injective in the 0-th cohomology.
2.8. We also need the following lemma (cf. [25], Lemma 4.1). Let N, Ng be closed C^-submanifolds of M with N=3No. Let ^=codim N, ^o==codim No. Let Y denote the complexiflcation of N in X.
LEMMA 2.5. — IfY is non characteristic/or M, then
H1 R r^xnT^x R ^om^ (^, ^oix) = 0 (i < d^
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and the natural homomorphism
(2.8) H^ R ^om^ (^, ^®^\M -^ H^o R r^x^x K -^^x (^ ^No|x)®or^|M
^ injective on T^ X 0 T^ X.
2.9. Let Q be an open subset of M with C^-boundary N and lying on one side of N. Let M be a coherent Q!^-modu\e and assume that N is non characteristic for M. Then we have an injectivity theorem ofmicrolocal restriction homomorphism:
PROPOSITION 2.6 [25]. — Lei Q.' be an open subset of M with ^-boundary contained in 0. and tangent to 0. at xeSQ. Then the homomorphism
P: H° R ^om^ (Ji, ^x) -^ H° R Jfom^ (^, ^x) is injective on n~1 (x).
This implies that the analytic wavefront set of the boundary value of a hyperfunction solution to M does not change on the fibre of x when one deforms the boundary (cf. also Kataoka [10], Lebeau [13]).
3. The theory of C^ | x (2)
This section is a supplement to te preceding section; we prove a basic property of the sheaf ^K i x ^or a C^-convex closed subset K of M.
3.1. Let M, X be as in Section 2. Let K be a closed C^-convex subset of M and let T^X denote the microsupport of C^. Let ^ K | X be the sheaf of microfunctions along T^X due to Schapira ([19], [20]): ^ | x= ^n (^hom (CK, ^x))®^ (c/- a^o Kataoka [7], [8], [9]; cf. Sect. 2.1).
Let xeK. Let (T^ M), = (T^ X),/(T^ X),; then (T^M),c=T^M. Let N be a 0°- submanifold of M with x e N such that
T ^ N = { z ; e T ^ M | < 9 , z ; > = 0 for all9e(T^M)^},
Y the complexification of N in X. For jpeT^XFU"1 OO, denote by ^p the fibre passing through p of the composition
(T^X),c,(TSX)^(T5YL.
The fibre ^fp is defined independently of a choice of N and called be p-leaf of T^X passing through p (cf. [26], Sect. 2.1). Note that ^p<^n~1 (x).
THEOREM 3.1. — Let K be a closed C^-convex subset ofM, xeK, and let u be a section
of^K\x\n-l(x) in a neighborhood of peT^Xr\K~1 (x). Let X be the p-leaf of T^X passing through p. Then supp(^) 0 ^f=^f in a neighborhood ofp ifu^O; i.e., u has the unique continuation property along JSf.
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To prove the theorem, we may assume that K is a closed convex subset of M = 1R". Let xeK. Let { ^ i , . . ., ^} be a set of linearly independent vectors of (T^M)^, and set L={x'eUn\(xf-x, ^ > ^ 0 (V7'e{l, . . ., d})}. Then we easily see that K<=L, and we have the following two lemmas:
LEMMA 3.1.1. — The natural ^ ^-homomorphism ^ i x IT? x "^L | x ls infective.
Sketch of Proof (cf. [19]). — By the trick of a dummy variable due to Kashiwara and for a choice of affine coordinate system (x^ . . ., x^) of IR", we may assume that M = IR", L = { x i ^ 0 , . . ., x^0](d<n), K = K ' x R^ for a closed convex subset K' of IR""1 and we have only to prove the injectivity at /?=(0; Q e T ^ X with Im^^O. Then, by using a quantized complex contact transformation (c/. [6], chap. 11), the problem is reduced to the following:
Let z = x + ( y ( = ( z i , . . ., z^)) be the affines coordinates ofC". Set
D ^ L e C l ^ ^ v f - ^ / . K r I D , = L e C " | ^ > S ( v , )2+ ^ yi I,
(. J = l J I .7=1 f e = d + l J
where d ( y , K') denotes the distance between y = (j^, . . ., y^_ i) and K/ in [R"~1. Define the sheaves ^Q^(k= 1, 2) on SD^ by ^^ = F^ (^c^/^c") 1^- Note that D^ cD^. Then
<9Di 0 OD^ corresponds to T^XriT^X, and the injectivity of ^ [ x l ^ x ^ ^ L I X ls
equivalent to that of the natural restriction map ^^ ^ ^ ^2 -^ ^02 I^DI n 502-
Let z=(z^, . . . , z^) be the affine coordinate system of C" which is real on IR",(z; 0 the associated coordinates of T* C".
LEMMA 3.1.2.—Let L = { x e [ R " | x i ^0, . . . , ^ ^ 0 } , and let u be a section of
^L | x \n~1 (OY Then u has the unique continuation property in the variables (£,1, . . ., ^) on { ^ = ( ^ , . . ., ^, D e T r - ^ l R e i ^ O ( z = l , . . ., d\ Rer=0}.
Sketch ofProof'. — We may assume that d<n, and we shall work on Im^^O. Set f ' 1
D = ^ z e C" | v, > ^ (^,)2_ + E ^2 ^ and ^ = 1^ (^c")/^c" ,D.
C j = l d<k<n }
Then, by using a quantized complex contact transformation, it is sufficient to prove that ^D has the unique continuation property in the variables (z^, . . . , z ^ ) on { z e 3D | ^i^O, . . . , ^ ^ 0 } - ^e ^all see this property in the variable z^ on { z e a D | ^ i ^ O , ^ . = ^0 (j=2, . . . . ^)} for fixed y] ^0(/'=2, . . . , r f ) . Let us set
f ' 1
D-= zEC"|^>(^)
2-+E(^-^)
2+ E ^ ^
I j = 2 d < f c < n J
and ^D' = ^D' (^c")/^c" I^D'- Since D' <= D, ^ |^ n BD' -^ ^D' I^D n 00' is injective; there- fore it is reduced to proving that ^^ has the unique continuation property in z^ on {ze9D' | ^i^O}, which is easy to see by using a local version of Bochner's tube theorem (cf. the proof of [18], Theorem 2.1).
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Proof of Theorem 3.1. — Let ^=(^, ^//) (= ^ + h"|) denote the fibre coordinates of T^X with ^eC^ ^eC"-^ such that { ^ = ( ^ , ^)eT^M |^==0} is the subspace of T? M generated by (T^ M)^. Then J^ = { { ; = (^/ +; r^, ; rT) | (^, 0) e (T^ M)^, V[ e ^} for some fixed r^eIR""^. For a set { ^ i , . . ., ^} of ^ linearly independent vectors of
f d 1
(T^M)^ we set P(^i, . . ., ^ ) = ^ = ^ ^.|^0(V/) ^ which is a polyhedral cone i j=i J
contained in (T^M)^. Let u be a section of ^ K i x n -1^ ) on an ^^ subset Uc:7T~1 (x) Pi T^X. Then it follows from Lemma 3.1.1 and 3.1.2 that the support of u is open (and closed) in U U {^(^r^ zrT)|(^ 0)eP(^, . . ., ^), r ^ e f f ^ } for any fixed T[" eV1 and for any linearly independent subset { ^i, . . ., ^} of (T^ M)^. Since (T^ M)^ is a ^-dimensional convex cone, this completes the proof
Remark. - In the case where K has C^-boundary, Theorem 3.1 was first proved by Kataoka ([7], [9]) (c/. [7], Prop. 1.8).
3.2. We give a few corollaries of Theorem 3.1. We shall refer to these results in Section 5.1 only in a very special case.
Let K be a closed C^-convex subset of M, Q = M\K.
PROPOSITION 3.2. — Let xeK and let Ji be a coherent Q) ^-module. Assume that there is a real-valued (^-function g on M mth g (x) = 0 satisfying:
(a-1) K c = { g ^ 0 } in a neighborhood of x;
(a-2) considering g a holomorphic function on X, (x; ^g(x))^Char(^).
Then we have: (i) ^om^M, ^|x) In-^^O, (ii) WR^om^^, ^|x)|n-1 (:c)=0 0-<0).
Proof. - Let p e T^ X 0 Char (^) U TT -1 (x). Let ^^ be the p-leaf of Tg X passing through p. Let Z = { z e X | g (z) = 0 }, p : T* X x ^ Z ^ T* Z the natural projection; then
^p ^ P ~1 P (P) n T^ X by the very definition of J^,, and p ~1 p (p) U T^ X ^ {7?} in every neighborhood of 7?. On the other hand, since Z is noncharacteristic for M, P ~1 P(^) Pi Char(^)={7?} in a neighborhood of/?. Let ^ e ^om^ (^, ^ | x)p? ^en supp (^) c= [p}. Hence it follows from Theorem 3.1 that u = 0. This completes the proof of (i). The second assertion follows immediately from the first one, by using the distinguished triangle (2.2).
PROPOSITION 3.3. — Let x, K, ^ be as in Proposition 3.2. Let u be a hyperfunction solution to M defined on M\K. If SS^f (u) 0 n~1 (x) <= { 0 }, then there exist a neighbor- hood U of x in M and analytic solutions u^ to ^i defined on U such that u^ |o = u on the component U^ o/U\K, where a indexes the set { U ^ } of the components of\J\K.
Proof. — Note that, since SS (u |n n u)c T^ M for a neighborhood \J of x, u is analytic on Q 0 U. By the assumption of existence of a C^-function g satisfying (a-1) and (a-2), the problem is reduced to the case K= [g^O}.
Let K = { ^ 0 } with dg^O. Let FeO^D^X)). Then we have (cf. [6] and [19]) R7i,|Lihom(C^, F)^R^fow(C^, Cx)®F^F(x)C^(x)orM|x[-^].
R7c^hom(C^, F)^Rj^m(C^, F)^RF^(F).
4eSERIE - TOME 25 - 1992 - N° 1
Hence we have the distinguished triangle
F| M ® C ^ ^ R ^ ^ ( F ) ( x ) o r M | x M ^ R ^ * ^ h o m ( C ^ F)(x)orM|xM -^ ,
where TC:T*X\X-^X. Putting F=Rjf6wz^(^, (9^) and taking the induced long exact sequence, by Proposition 3.2(ii), we get an exact sequence
0 ^ ^om^ (^, ^M)®C^ ^ ^om^ (^, ^i) ^ ^ H° R ^m^ (^, ^, x).
This completes the proof.
4. A preliminary theorem
4.1. NOTATIONS. — Let M, X be as in Section 2. Let Q = { ( p > 0 } be an open subset of M with analytic boundary N = { c p = 0} (^(p 7^ 0). Let XQ e N.
Let ((pi, (p^) be a pair of real-valued OMunctions on M with (pi (^o) = (p^ (x) = 0,
^(pi A^cp^T^O. Let Q/ := { q ) l > 0 , (p2>^} an(^ assume Q'czQ.
Set N Q = { X G M |cpi (x)=(p2(x)=0}; then XoeNoC^Q'. Let Yo(resp. Y) denote the complexification of No (resp. of N) in X,
p o : T * X x ^ Y o - ^ T * Y o , p : T * X x ^ Y ^ T * Y the natural projections.
4.2. STATEMENT OF THEOREM. — Let Q, Q' be as in 4.1.
Let ;?eT^X PiTi"1 (xo). Let / be a homogeneous holomorphic function on T*X defined in a neighborhood of p with /(/?)==0. Let M be a coherent J^x"
module. Assume the following:
(4.1) Yis non characteristic for^;
(4.2) Char (^) c= {/ = 0} in a neighborhood of p\
(4.3) {/,(p}(/^0,
with { ^, ^ } denoting the canonical Poisson bracket on T*X.
Let Cp = [ q e RQ 1 po (p) D T$ X \f (q) = 0}; then Cp is a real nonsingular curve passing through p and contained in po"1 po (7?).
THEOREM 4.1. — Assume (4.1), (4.2) and (4.3). L^ U ^ an open connected interval ofCp on which {/, c p j ^ O . T/?^7Z the homomorphism [c/. (2.3)]
H0 R r^ (U, R ^om^^ (^, ^, x) | C^) ^ H° R F^ (U, R ^m^ (^, ^/, x) I C,) z^ injective for any closed proper subset Z of\J.
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE S U P E R I E U R R
4.3. PROOF. — Since there is an open subset Q ^ c Q with analytic boundary N^ with No<=Ni, on account of Proposition 2.6, we may replace Q with Q^ and assume from the beginning that NQC:N. Let us then choose a system of local coordinates x=(xi, . . ., x^eV o f M so that Q = = { x i > 0 } , N o = { x i = X 2 = 0 } .
Let z=x-\-iy be the complexiflcation of x, (z; 0=(zi, z^, z"\ ^, ^ £,") (^ == ^ +; r() be the associated coordinates of T* X. We may assume that N^ (Q') c= Q' x (— G) for a closed convex proper cone G of !R2 with { ^ ^ 0, ^2 = 0}c: ^ therefore, by Corollary 1.4, we have
T ^ X c { ^ 0 , ^=0, ^"=0}, T^,Xc={(^, ^)eG, ^"=0}.
We then have the following commutative diagram [cf. (2.3) and (2.4)]:
R 3^'oma., (J^, %'n | x)^11 ^ow^., (^, %'n-1 x)
bv bvI I
i 5 -I
R ^{ ^ 1 ^ 0 ^ 2 = 0 }F PI ^ R ^{ ( ^ ^ 2 ) e G }F P].
where F=Rc^ow^(^, ^ N o i x ^ ^ N o l M - The left vertical arrow bv is factored as follows (c/. Sect. 2.3):
R ^m^ (^, ^ i x) ^ R r^ x n T^ x R ^^^x (^' ^N i x)®o^ i M [1]
4 R r^ x n Tn X R -^W^x (^' ^No I x)®OrN, |M[2].
Here H°(^) is injective by Proposition 2.4, and so is H0^) by Lemma 2.5; there- fore, since WRj^om^^, ^Q|x)=° O'<0). the map H^Rr^U, b^ °b^ |c ) is also injective. Hence it is sufficient to prove the inject! vity of the map
^Rr^u.sl^iHKu.Rr^^o^^o^^-^HK^RroF).
Let us take the long exact sequence induced from 5:
. . . ^ H^(U, Rro^^F) ^ Hj(U, Rr^o,^o}F) ^ Hi(U, Rr^F) ^ . . . We shall prove that the first term is trivial [L^.,H^(U, (RrG^^=o}F)|c )=0]. Since HiF = 0 (i< 1) by Proposition 2.3, we have
Hi(U, (RF^^o}F)|c,)=rz(U; (F^.^^H^)^)
= l l m o ^ u { ^ £ ^ ( O n { ^ ^ 0 } ; HlF ) | s u p p ( ^ ) c = G , s u p p ( ^ ) n U c = Z } , where 0 runs through the family of open neighborhoods of U in T* X C\ {^\= 0} and supp(^) denotes the closure of supp(M) in 0. Take a section u. Suppose that supp(^) 0 U^0, and we shall see the contradiction. Since Z ^ U and U is connected,
4eSERIE - TOME 25 - 1992 - N° 1
there is a boundary point ^=(Ci; q ' ) (^=p(^)eT^Y) of supp(i<)nU in U. Con- dition (4.3) implying 5//5^(^)^0, there exist e>0, an open neighborhood 0' of q' in T* Y and a homogeneous holomorphic function ^ on 0' such that
OeO {/=o}={^=g(z', o, (z-, oecy}
with 0 , = { z i = 0 , | ^ - C i | < £ } x O \ Set
o^o.n^^o}, o^^cyn^^o}.
Then p | o ^ ± ) ^ ^ = o ^ gives a holomorphic isomorphism from .0[±)(^[f=0] onto (V^. Thus it follows the division theorem of microdifferential operators for micro- functions with holomorphic parameters (cf. [3], [5], [15]) that there is an isomorphism (4.4) (p |o^±))^ (H1 F |o^)) ^ ^om^ (^±), <^o l v),
where ^^^(plO^^^y -x®^^!0^^ which is a coherent ^Y^odule on (V^. Since every section of ^N()|Y has the variable ^ as a holomorphic parameter (c/- [3]. [5]^ [15]), (4.4) implies that every section of H^jo^) has the variable ^ as a holomorphic parameter on the set O^ P| T^X 0 {/=0}.
Since q is a boundary point of supp (u) C\ U, there is a sequence { ^ ; v e I^J}
(^(^ ^^ ^^i) of ^ converging to q with ^supp(^). Then, perturbing ir^ of
^, we get a sequence { ^± )} (^^(^'^ ^'v), ^//)) of p o1 po(^) n { ± ^ > 0 } (for each ±) so that ^±) -> q, f (^±)) == 0, q^t supp (^). Hence, by the unique continua- tion property in the variable ^ °^ ^e sheaf H1 F|0^± ) 0 {/=0}, we have ^ = 0 on O ^ n T ^ X (by shrinking Og if necessary), which is a contradiction. This completes the proof.
5. The complex C^ | x for the exterior domain Q of a corner
Henceforth we shall work on a fixed real analytic manifold M of dimension n and use the notations prepared in Section 2.
5.1. NOTATIONS. - Let QI, Q^ be open subsets of M with 0°-boundary N^, N^
respectively. Suppose that N^ and N^ intersect transversally. We set Q = Q i U Q 2 .
(5.1) K=M\Q, No=NiHN2
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE
(cf. Fig. 5.1); K is a C^-convex closed subset of M with piecewise smooth boundary having a corner along No. Let Y^, Y^ be the complexification in X o f N ^ , N^ respectively;
setYo=Y,HY,.
__ N.
Fig. 5.1.
5.2. Let Q=^i U ^2 be as in 5.1. Let T^X denote the microsupport of CQ, and let
^n I x ^e tne complex of microlocalization of (9^ along T^X (c/. Sect. 2.1):
^ I x = ahom (C^, ^x)®orM M.
PROPOSITION 5.1 (Schapira [21]). — The sheaf H° (^ | x) ^ supported in T^X, W ^ homomorphism [cf. (2.3)]
(5.2) H° (^ , x) ^ H° (^ , x)©H° (^, | x) is injective.
Sketch of Proof. — The first part is already proved in Corollary 2.2. Let K^=M\H.
O'=l, 2); then it follows from Proposition 2.1 that ^ K I X J T M X ^d ^Ki\x\-r^x O^L 2) are subsheaves of ^^. It is sufficient for the second part to prove that, at peT^X, (5.3) ^ K | x ^ = ^ K , i x ^ n ^ K 2 | x , p in ^
By using a quantized complex contact transformation (cf. [6], chap. 11), the problem is reduced to the following:
Let z=x-\-iy(=(z^ . . ., z^)) be the affine coordinates of C". Set Do=LeC" Yn>^y]\
I j-i J
D ^ L e C - ^ O ^ + ^ l ,
I J = 3 J
D ^ L e C - I ^ O ^ + ^ l ,
I J - 3 J
f "- 1 1
D 3 = ^ z e Cn| ^ > ( ^ )2_ + ( ^ )2+ ^ ^ l ,
I J - 3 J
4eSERIE - TOME 25 - 1992 - N° I
and define the sheaves ^ o11 ^ (0^^3) by ^= r^ = F^. (^c")/^ \ODI. ^en (5.3) is equivalent to the equality
(5.4) ^(5D350=^aDl?0 H^D2'0 "^^DO'O-
Since 03 is the convex hull of D^ UDi? ^is follows from a local version of Bochner's tube theorem (c/. e. g. [4], Prop. 3.8.6). •
Let ^ be a coherent ^ ^-module defined on an open subset of M. The following proposition is a special case of Proposition 3.2.
PROPOSITION 5.2. — Assume that Y^ and Y^ are non characteristic for M'. Then we have H1 R ^om^ {M, ^ | x) = 0 (;< 0).
COROLLARY 5.3. — Assume that Y^ and Y^ are non characteristic for ^i. Let u be a hyper function solution to Ji on Q. If SS^f (u) c= T^ X, ^^ ^£^^(0).
Let ^3 be another open subset with C^-boundary N3 such that 030=0 and No €=N3. Note that T^X (^n~1 (No)c=T^X. Let Y3 be the complexification of N3 in X. Then we have:
PROPOSITION 5.4. — 1/^3 is non characteristic/or M, then the homomorphism H° R ^om^ (^, ^, x)p ^ H° R ^om^ (^, ^ , x)p
^ injective at p e T^ X\T^ X.
Proof. — Let us choose a system of local coordinates x= (x^, . . ., x^) e IR" of M such that 0 ^ . = { ^ > 0 } (f= 1, 2). Let z be the complexification of x, (z, 0=(x+(y, ^+;'T|) the associated coordinate system of T* X. We set
AO={(X+Z>, ^+zn)|^=^=o^=o, ^ < o , ^ < o , ^ = . . . -^-o},
A,=[(x+iy, ^+/r|)[xi=0,x^0^=0, ^<0,^=^= . . . =^=0},
A,-={(x+^, ^^)|x^0,^=0^=0, ^<0,^=^= . . . =^=0};
then T^ X\T^ X = Ao U Af U A^-.
(Ca^ l)j9eAo. — Since SS(C^\No) H Ao=0, C^ and C^ are isomorphic in D^X; 7?);
therefore the ^x"homomorphism ^01 x "^ ^K [ x ls an isomorphism in a neighborhood of j9 \cf. (1.3)]. On the other hand, since ^^T^X, we have an isomorphism
^Q[X ^ ^ K I X [1] m a neighborhood of p [cf. (2.2)]. Thus, putting K3==M\Q3, we have a commutative diagram
H° R ^om^ (^, ^ (x)p ^H° R ^om^ (^, ^,, x)p
(II Ml
H1 R^om^ (^, ^ (x)p -H1 R ^om^ (^, ^, , ^
^ 1 1 ( I I
H1 R ^f^m^ (jr, ^o | x),^H1 R ^om^ (^, ^,, ^.
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUP'ERIEURE
Let p3 : T* X x x Y3 ^ T* Y3. Let M^ = ^3 _ x,p ®^ (^x ®^A. then ^} is an
^Ys'psCp)"1110^11^ °^ ^lmte type• By tne division theorem of microdifferential operators for microfunctions with holomorphic parameters (c/. [3], [5], [15]), we have a commutative diagram
H1 R ^om^ (^, ^No | x)p ^H1 R J^w^ (^, ^31 x)p ( I I ( I I
Hom,^ (^, ^ , ^, „ (,))- Hom^ (^, <^ „ ^),
Hence the injectivity to be proved is equivalent to that of the last horizontal arrow, which is a conclusion of Proposition 2.1 since p3(/?)eTg ¥3.
(Case 2) peA^ U A ^ . - Let p e A ^ . Since N3 is tangent to N^, by replacing ^3 with a smaller one, we may assume that 030=0^. Then we have the decomposition of the morphism P = Pi ° ^^'.
H° R ^om^ {M, ^ | x) •^H° R ^om^ (^, (^ , x) i^ ^
H°Rjfom^(^,^|x). 1
Set N ^ N i U K . Since SS(CK^NI-) nAi" =0, CK and CN,- are isomorphic in D^ (X; 7?); therefore ^r l x ^ ^K | x ln a neighborhood of p. Thus, in the same way as in Case 1, the injectivity of P^ is reduced to that of the homomorphism of sheaves on T^Yi :^Nf |YI ITN Yi -^Nr On the other hand, it follows from Proposition 2.6 that Pi in inject! ve, which achieves the proof.
5.3. THE CASE WHERE YQ IS NON CHARACTERISTIC.
PROPOSITION 5.5. — Let M be a coherent 2^-module. If Y() is non characteristic for Ji^ then we have
(5.5) H° R ^om^ (^, ^, x) |.-1 (NO)\TM x = 0.
Proof. - We choose a local coordinate system as in the proof of Proposition 5.4 and use the same notations. Let /?eAo. Then we have the isomorphism
(5.6) H° R ^om^ (^, ^, ^ ^ ^xt^ (^, ^, ^.
Let p o : Y o X x T * X ^ T * Y o . Let ^=^ ^x,p®^x ®^M\. Then, since YQ is noncharacteristic for M, M^ is an ^y^^^^-mod^e of finite type and the division theorem of microdifferential operators gives the isomorphism
/< 7\ 'R'vtJ ( //^ (/^ ^ / ^ / T 7 v + J—2 / ^/{ p } (/? \
^ • ^ ^X p <^ ^No | X p) = Ext^ p, (p) Wo, ^NO po (P)Y
(5.6) and (5.7) prove that H° R J^om^ (^, ^, x)p = 0.
4^^^ - TOME 25 - 1992 - N° 1
Let ^eAi-UTT^No). Let p, : T * X X ^ -^T*Yi, ^=pi(^). Let ^} be the (^Y^q-module of finite type induced from (^x®@x^)p- ^et ^i^^i ^ K - Then we have (cf. the proof of Proposition 5.4)
H° R ^om^ (^f, ^, ^ ^ Ext^ ^ (^,, ^f | x p) ^ Hom^ ^ (^}, ^f | Y,,).
Since YQ is non characteristic for M, YQ c^ Y^ is non characteristic for M^^ at ^. Hence, by the unique continuation property of the sheaf ^f |YI (^/' PL Prop. 1.8), we have Hom^ (^Y^? ^Nf I Yi q ) ^ ^ ' This completes the proof.
In particular, when M is an elliptic system, we have the following theorem of Bochner type as an immediate corollary of Proposition 5.5 and Corollary 5.3.
COROLLARY 5.6. — Lei Q=Qi U^2 ^e as m 5 - 1 - Let YQ denote the complexification of the corner Ng of M\Q. Assume that M is elliptic and that Yg is non characteristic for M. Then the restriction homomorphism
^om^ (^f, J^M) INQ -^ ^ ^o^^ (^ ^M) INO is an isomorphism.
Remark. — The boundary value problems for elliptic systems- are systematically studied by Kashiwara-Kawai [3], and Corollary 5.6 is just one of the conclusions from their main theorem. But here we are stressing the fact that the noncharacteristic condition of YQ makes the problem trivial outside T^ X.
6. Microlocal analysis at the corner of an obstacle
Let Q=QI U ^2 ^e a domain given as the union of two open subsets with transversal C^-boundaries (cf. 5.1). We follow the notations of 5.1. For pe^o x ^ T ^ X , we set
^p = Po 1 Po (?) l^ T^ X, with po being the natural projection Y() x x T* X -» T* Yo.
6.1. A KEY LEMMA. — Let p G N() x ^ T ^ X . Let/be a homogeneous holomorphic function on T*X defined in a neighborhood ofp with/(p)==0.
Let ^i be a coherent ^^-module. Assume the following:
(6.1) each ofY^ andY^ is non characteristic for^;
(6.2) Char (J^) c= {/ = 0} in a neighborhood of/?;
(6.3) Im/|T^X=0;
(6.4) {/,cpi}(7^0, {/,(p2}WO,
where (pi, cp2 are defining functions o f Y ^ , Y^ respectively with d^)^ A rfep^^O.
Set Cp=EpD[f=0}; then Cp is a nonsingular curve in Ep passing through p, in which curve the sheaf H° R J^fom^ (M, ^ (x) IE ls supported.
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE
LEMMA 6.1. — Assume (6.1)-(6.4). Z^/ U be a small connected open neighborhood of p in Cp, Z a closed proper subset of\J. Then the homomorphism [cf. (2.5)]
Fz (U, H° R ^om^ (^, <^ i x) | C,) -. F^ (U, H° R ^om^ (^f, ^., x) | C,)
is injective (/= 1, 2).
Proof. — We shall prove the injectivity for 7=!. Set Qi^Q^FiQ^ ^d define the complexes of sheaves: F = R J^om^ (e^, V^ \ x)? F\ = R ^om^ (^,^^) x) (fe==l, 2, 12), Since F^, F^, and F are concentrated in degrees^O in virtue of (6.1), H°Rrz(U, Fk)-^rz(U, H°Ffc) is an isomorphism for k= 1,2,0. Thus we have the commutative diagram
rz(U,H°F) ^ rz(U,H°F,)
?2 I I P l 2 / l
rz(U,H°F2) -^ H°RFz(U,F^).
Pl2/2
Let Merz(U,H°F), Pi(^)-0. Then (3^/2° P 2 ^ ) = P i 2 / i0 Pi (^)=0- Since P^/2 is injective by Theorem 4.1, we have ^^(u)=0. Together with P^ (^)=0, this implies u=0 on account of Proposition 5.1.
Remark. — The first (resp. second) condition of (6.4) on/is not necessary to prove injectivity of the restriction homomorphism from Q to Q^ (resp. Q^)-
In particular we have
THEOREM 6.2. — Assume (6.1)-(6.4). Then
F^ ^ (H° R ^om^ (^, ^ i x) | E^) ^ F^ ^ (H° R ^f^m^^ (^, ^., x) | E,)
is injective (7= 1, 2).
Proof. — Since /? is not an isolated point of Ep n { / = 0}, we can apply Theorem 6.1 for a small connected open neighborhood U of p and Z = { p } .
NOTATIONS (c/. Sect. 2.6). — Let M be section of H°Rj'fow^(^, ^ o j x ) - I11 ^is paper, we use the following notations for simplicity: SS^ (u) == supp (u) ^ T^ X and SS^. (M) = SS^ (M) U T^ X 0- = 1, 2). (Cf. also Sect. 2 . 4 and 2.5.)
COROLLARY 6.3. — Assume (6.1)-(6.4).
Let u be a section of H° R ^om^ (^, ^Q | x) ^ a neighborhood of p. If SS^ (u) r\ Ep= {j9 } ^ a neighborhood of p, then pe SS^ (^).
Proof. — Suppose that /?^SS^(M), and we shall see the contradiction. Since q^SS^(u) for every qeEp close to p with ^7^, it follows from Proposition 5.1 that q^SS^(u). This implies that u belongs to the stalk of the subsheaf r^(H°R^m^Cjr, ^nix)^)- Then, by Theorem 6.2, we have u=0, which is a contradiction.
4eSERIE - TOME 25 - 1992 - N° 1
6.2. PROPAGATION BEYOND A CORNER OF ANALYTIC SINGULARITIES. - We apply Corollary 6.3 to the case where the bicharacteristic curve of/passing through p is transversal to a corner.
Let pe^Q x^T^X. Let / be a homogeneous holomorphic function defined in a neighborhood ofp with/(^)=0. Let Ji be a coherent ^ ^-module and assume (6.1)- (6.4). Denote by H^ the real Hamiltonian vector field of g == Im / on (T* X, Re co). We assume moreover
(6-5) ±^tH^(^C^(K;No),
where dn: T^T*X-^T^X and C(K;No) is the normal cone of K along No (cf. 1.1). Let b±(p) denote the positive (resp. negative) integral curve of H^ issued from p. Condition (6.5) implies that, roughly speaking, b± (p) issues into Q transversally to the corner K (cf. Fig. 6.1).
b-(p)
Fig. 6.1.
NOTATION. - For a section u of H° R ^om^ (Jl, ^, x), let SS^lQ^SS^nn-^O).
Remark. - When u is a hyperfunction solution to M defined on 0, SS (u \ Q) is nothing but the singular spectrum (i. e. the analytic wavefront set) of u over Q.
THEOREM 6.4.—Assume (6.5) in addition to (6.1)-(6.4). Let u be a section of H° R Jfow^ (^, ^) x) ^ a neighborhood of p . If b ~ (p) cz SS (u | Q) and if b~(q)(^SS(u\Q)^0 for neighborhood Cp(s) ofp in Cp and for every qeCp(£)\{p], then p e SS^ (u) and b + (p) cz SS (u [ Q).
Proo/. - This is an interpretation of Corollary 6.3 by using the theorem of propagation of regularity up to the smooth boundaries N^, N^ (cf. Kataoka [7], Schapira [16], [17]).
This theorem asserts that an "isolated" singularity p of u propagates up to the corner and goes beyond the corner along the bicharacteristic curve b (p).
ANNALES SCIENT1FIQUES DE L'ECOLE NORMALE SUPERIEURE
6.3. DIFFRACTION BY A CORNER. - Let ? £ NQ x ^ T ^ X . Let/, ^ and ^(j?) be as in 6.2. Note that n^ (j?))c=0^ (resp. Q^) if and only ifn(b~ (/?))c=Q^(resp. Q^).
As a corollary of Lemma 6.1, we have the following theorem which proves the appearance of diffracted rays at the corner of K (cf. Fig. 6.2).
SS(u|Q)nb""(p)=0
and peSS^(u)
Set again C ^ = E ^ U { / = 0 } .
THEOREM 6 . 5 . — L e t M be a coherent Q^ ^-module satisfying the conditions of 6.2 at p. Let u be a section of R ^fom^ (^, ^Q | x) m ^ neighborhood of p. If p e SSo (M^SS (u Q) P| b (/?), then there is a neighborhood C (c) of p in C such that Cp (8) c= SS^ (u) and b + (q) c SS (u \ Q) for all q e C^ (s).
Proof. — We may assume that TI;(Z?~ (7?))c:Q^. Assume that there exists a sequence { ^ } of Cp converging to p such that ^+(^)cj=SS(M|Q). Then, by propagation of regularity up to the boundary N^, q^^SS^(u). Since on the other hand/? ^ SS^ (^), it follows from Proposition 5.1 that q^^SS^(u) for v ^ > l . Thus, by Lemma 6.1, p^SS^ (u) implies that p ^ SS^ (u), which is a contradiction.
COROLLARY 6 . 6 . — L e t u be as in Theorem 6.5. If b ~ ( p ) c): SS (u \ Q) and if there exists a sequence [ p ^ ] ofSS^(u) converging to p , then b+ (q) cz SS (u \ Q) for all ^eCp(s), with Cp (s) being a neighborhood of p .
Remark. — b+(q) (qeCp(&)) are the so-called diffracted rays produced by p e SS^ (u)\SS (u | Q) U b ~ (p) at the corner (cf. Keller [11]).
6 . 4. BOUNDARY ANALYTIC SINGULARITIES IN THE COMPLEX REGION.
Let Q = Q i U ^ 2 - Let ^ " " { ^ s ^ } be an open subset of M with C^-boundary N s = { ( p 3 = 0 } ?3^0) and contained in Q. Let Y 3 = { z e X | ( p 3 ( z ) = 0 } .
Let x e No Pi N3, p e T^ X Ft TT" 1 (x). Let / be a homogeneous holomorphic function on T*X defined in a neighborhood ofp with/Cp)=0. Assume
(6.6) { / , < P 3 } ( ^ ) ^ 0 . 4eSERIE - TOME 25 - 1992 - N°
Let po : T* X x x Yo -^ T* Y() be the natural projection. We then set (6.7) CW= { ^ e T S 3 X n 7 l -l( x ) | / ( ^ ) = 0 , p o ( ^ ) = p o ( ^ ) } ;
Cy is a nonsingular curve in a neighborhood of/? passing through p, since {/, (pa } T^O.
Let M be a coherent ^x'1110^!^ defined in a neighorhood of x and assume that ¥3 is non characteristic for J(. Then we have:
THEOREM 6.7. — Let p^T^X. Let/be as above. Let Ji be a coherent 2 ^-module for which Y-^ is noncharacteristic. Assume that Char (M) c { f= 0} in a neighborhood of p. Let u be a section of H° R ^om^ (y^, ^ n ) x) w a neighborhood of p. Then
(i) supp (u) n po' po (P) n TS, x c: c^.
(ii) C^crSuppO^) ^ a neighborhood ofp ifu^O.
Proof. — The first assertion is trivial. To prove the second assertion, by replacing Q = = O i U ^ 2 by 03 U ^2 (or ^lU^s), we may assume from the beginning that 03 = QI . Define C (= C^) by (6.7).
Let us define the complexes of sheaves F^, F^, F^, and F on T*X as in the proof of Lemma 6.1. Let Z be a closed proper subset of a small connected neighborhood of p in C and consider the commutative diagram
H° R Fz (C, F) ^ H° R Fz (C, Fi)
?2 | I P l 2 / l
H°RFz(C,F2) ^ H°RFz(C,F^).
P l 2 / 2
Since T ^ X H T ^ X C = T ^ X , F^O on C in a neighborhood of p', thus P i 2 / i ° P i
= P ^ ^ ° P ^ ls tne zero map. Then, since P^/i is injective by Theorem 4.1, Pi is still the zero map. The complexes F and Fi being concentrated in degrees ^0 by Proposition 3.2, we have
H° R Fz (C, F) = Fz (C, H° F) and H° R F^ (C, FQ = F^ (C, H° F^).
Thus the map
r^c.H^-.r^c.H^)
is trivial. On the other hand, this map is injective by Proposition 5.4. Hence we have Fz(C, H°F)==0, which completes the proof.
Remark. — The above theorem will be applied in the forthcoming paper to the problem of continuation of analytic solutions to differential equations.
7. Application to Dirichlet problems
Let QI = {(pi >0}, 0^ = {q>2 >0} be a pair of open subsets of M given by C^-functions (pi and (p2 with <Api A rfq^ 7^0; let 0=0^ U ^2 (c/ ^ • 1)' ^e follow the notations of 5.1.
ANNALES SCIENTIFIQUES DE L'ECOLE N O R M A L E SUPERIEURE
7.1. ASSUMPTIONS AND NOTATIONS. - Let P=P(x, D) be a second order differential operator with analytic coefficients with principal symbol /. We shall always assume that each of Y^ and Y^ is non characteristic for P and
(7.1) Im/|TSX=0.
Let pj : T*X x ^Yj-> T*Y^. (/'=0, 1,2) be the natural projection.
Let ^eT^Yo; set E (qo) = p o1 (qo) 0 T^ X, which is an affme plane in T^ X 0 TI; ~1 (TI; (^o))- Considering that/1 E (qo) is a polynomial of degree two, we assume moreover that
(7.2) C = [p e E (qo) \f(p)=0] is an ellipse in E (qo).
Henceforth we fix qo throughout.
Let H^ denote the real Hamiltonian vector field o f ^ = I m / o n (T*X, Re®), and let Z?1 (p) denote the positive (resp. negative) integral curve of H^ issued from peC. Set
C^={peC\±^(p)^^>0}, D , = { ^ e C | < H ^ ) , d ( p , > = 0 }
for k ==1, 2. C is divided into four regions C°[ H C|(a= ± , (3= ±) by four points of DI U D ^ (cf. 7^. 7.1). Every point of Ci- U C^-(resp. C^ U C^^) corresponds to an incoming (resp. outgoing) singularity at the corner on E(^o); i.e., if peC"[ U C^ (oc= ±), then b^ (p) is transversal to K and n (b^ (p)) c Q.
For ^eC, we denote by p^ (^=1,2) the reflected point of p with respect to N^, L^., the point of C satisfying p^1 ^^(p) C\ C={p, p^] (cf. Fig. 7.2). Observe that D ^ = { 7 ? e C p=p(k)] and^eC^ if and only if ^eC^ (^= 1, 2).
772 -axis
^2 ^(P)
Pl"^!^)
Fig. 7.1. - C = E ( ^ ) n ! . / ==O j . Fig. 7.2.
7.2. BOUNDARY CONDITIONS. - Set 8jfi. = 80. P| N^\No (7= 1, 2); ^. Q is an open subset of Ny lying on one side of No (c:N^.), and 80.= S^ Q U 9^ Q U No (disjoint).
4eSERIE - TOME 25 - 1992 - N° 1
7.2.1. Boundary values on o^Q. and S^O. - Let P be a differential operator of 7.1. Let ^ = ^ x / ^ x P = ^ x ^ w i t h u=\ mod ^P-
The tangential system M^^ of ^ on Y^ is given by definition as
^Yi=^Yi-.x®^x^
^Yi OYI -> X®^) 0 ^YI OYI - X ® ^ ^
where ^vi=S^Pi/^.3/^. for a choice of local coordinates (x^ . . ., x^). Define a coherent ^y^-submodule J/^ of ^y^ by
^\=^(4^x®^).
We have the following chain of C-linear maps:
(7.3) r (E (^o); H° R ^om^ (^f, ^ , ^ r (E (q^ H° R Jf^z^ (^, <^, „))
^ r(p, (E(^)); jfow^ (^, ^))
bv^ r(pi (E(^)); ^f^m^^ (^i, ^i))
^ r (pi (E (^)); H° R ^f^^^ (^,, ^^ ^ , y^)) [c/. (2.3) for the first and the fourth maps; cf. (2.4) for the second map]. For a section u of H°R^fow^(^, ^|x) on E(^o), we denote by u\^^ the image of u in r (pi (E (^o)); H° R ^om^^ (J^i, ^^ | Yi)) by the composite of (7.3).
Remark. - Let u be a hyperfunction solution to P defined on 0. Denote by u \ ^ the first trace of u\^ on N^ as a hyperfunction (cf. [8], [10], [12], [16], [17]); se[
u laic == (u | N,) |^Q, the first trace of u on Bi 0. Then M |^ =^(u 5^), where
a : r(^Q; H°Rjf^^(^,, ^))-^r(p,(E(^o)); H°R^^^(^,, ^^,^)) (cf. Sect. 2.5). In particular, SS,^ (^ ,^) U pi (E (^)) = 0 if and only if u |^ = 0 on Pi(E(^o)).
We also define a section u\^ of H°R^W^(^ ^"IY^) through the same procedure as above for J^ = ^Ys (^2 - x®^)-
7.2.2. ^ /^wwa. - We follow the notations of 7 . 1 and 7.2.1. For a section u of R ^om^ (^, ^ | x) on E (^o), we denote by M [^ the image of M by the first map of (7.3); we set SS^ (u) = supp (^ |^) (c/. Sect. 2.6).
LEMMA 7 . 1 . — Let u e r (E (^o); H° R .Tom^ (^, ^, „)) w^ M | ^ = 0.
(i) L^ U-^ be an open subset ofC^, set U-=pi-1 p l ( U+) n C ^ . ^^w^ ^ | ^ = 0 on
u +- ^ ^ k = 0 ^ a point p^eV, then u\^=0 on L^. (U-" ^ ^^ necessarily connected.)
In particular we have the following:
(ii) Let peC. I f p e SS^ (^) and if there is a sequence [p^} of C converging to p with p, i SS^ {u\ then p^ e SS^ (^).
ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE
Proof. - We shall prove (i). Let v = bv (u \ o^) be the image of u |^ by the second map of (7.3), w the image ofv by the third map; then wer(pi(E(^o)); ^w^(J^\, ^N1)).
and u |^Q is the image of w by the fourth map.
Assume that u\^=0 at a point p^ of V. Since u\^=0 at (^o'^DeU", we have w = 0 at pi(^). Let Gi=Ni\3iQ; which is a closed subset of N^ with C^-boundary No. We then have the short exact sequence
0 -^ Jfow^ (^i, ^Gi | Yi) -> ^om^^ (^i, ^N1) ^ H° R Jfwn^ (^\, ^, ^) [c/. (2.2)]. Hence it follows from the assumption (^1^=0) that w is in fact a section of the subsheaf ^om^ (J^, ^ | y^). The sheaf ^ , y^ having the unique continua- tion property on pi(E(^o)) (cf. [7], Prop. 1.8; [9], Prop. 4.1.11), we have w = 0 on pi(E(^o)).
Now let^eL^ with piO^)^!. Since ^Yi^i ("^Yi^i®^ ^Yi) is isomorphic to ^i^®(^Yi -x®^x^~) as an ^Yi^i-"10^1^
(7.4) ^ H ° R H o m ^ ^ ( ^ ± , ^ | x , p ± )
-^ Hom^^ ^ (^i „, ^, ^)®H° R Hom^ ^ (^-, ^, | x,,-), ((^ ni)?-^. (^|Qi)p-)^(^i. (^IniV)
is injective. Since M |^ = 0 on U ~ and w = 0 on pi (E (^o)), the injectivity of (7.4) implies that u \^ = 0 at any point p^ e L^. This completes the proof.
7.3. DIFFRACTION BY A CORNER OF INCIDENT RAYS. — As an application of Theorems 6.4 (or 6.3) and 6.5, we prove that a cone of diffracted rays is produced when a single ray hits the corner or when finitely many rays hit the corner simultaneously.
7.3.1. Let ^oeE(^o). Let (r(p r[^)eR2 be affine coordinates on E(^o) such that Oh? ^2) corresponds to the point /?=A)+^I Api+^'r^^ Then each fibre of Pfe | E too) (fe= 1, 2) is given by
pjE(^o) : (r|i, r|2)i->r|2, P2\E^o) '• Oil. ^i-^i and C is given by
p.,) c-{(n,.^) ^Y+2(''^Y
n^)c..e+(^)
2-ll
I \ a } \ a }\ b } \ b ) J for some <3>0, ^>0, 0<9<7i, (rii, r^eR2 (cf. F^. 7.1). Note that 6 is determined independently of the choice of (pi, (p^ and PQ.
7.3.2. Statement of Theorems. - Let M be a section of H° R Hom^ (^x/^x p. ^n | x) in a neighborhood of C (or a hyperfunction solution to P defined on Q). Assume that u satisfies the microlocal boundary condition
(7.6) 4^=0 onpi(C) and u\^=0 onp^C).
4eSERlE - TOME 25 - 1992 - N° 1
Remark. - In the case where u is a hyperfunction solution to P on D, (7.6) is fulfilled in particular if u satisfies the boundary condition
(7.7) "kn^i and u\g^=82
for g, e T (57D; .a^i) and ^ e F (^ 0; .^).
Cf. the notations in Corollary 6.3 and Theorem 6.4 for SS^ (u), SS^ (u), SS(i<|n) For ^ e C, let R, = [p, p^, p^, p^, p^ p^ ^ ^ p^ ^ ^ , , _ y
THEOREM 7.2. — Let u be as above. Let p e Cf U C^. Assume
Rpn(cruc2-)^}.
TACT! we have:
(1) ^ SS^ («) n C,- = { p } 0 €„ for k = 1, 2, fAen C ^ <= SS^, (») aw^ ^ c= SSn, («).
/« o/Aer worA, we have:
(ii) // ^ - (p) c SS (M 112) a«rf ;/ A - (p') ^SS(u\ Q) (V^' e Ci- U C;- w;^ D' ^.p) then
^ (^ c SS (u | Q) /or a// q e Cf U C^.
.Reward 1. - Let C be given by (7.5). Assume R^O (C^ UC,-) = { / ? } . Then Q and p satisfy one of the following (cf. Fig. 7.3):
(a) 9 = 7i/3, p=(f],±a/^3,^± 6/^/3).
(b) e=TC/2,j9=(r|i±a,r|2),(rii,n2±A).
(a) e = 7 t / 3
(b) 9 = 7 T / 2Fig. 7.3.
7?ewa^ 2. - The positive ^characteristic curves b+ (q) (qeC^-UC!) are called the diffracted rays produced by a single incident ray b- (p), and they form the surface of a cone (cf. Fig. 7.4; cf. Keller [II], Fig. 5).
ANNALES SCIENTIFIQUES DE L'ECOLI: N O R M A L E S U P E R I E U R E
Fig. 7.4. — Diffraction of a single incident ray.
diffracted rays
THEOREM 7.3. — Let u be as above. Let Z be a finite subset ofC~^ U C ^ . Assume that there exists peT. such that Rp 0 (C^ U C^)4=Z. 77^2 we have:
(i) // SS^ {u) U C,- == Z U C,- fork=\,l, then C^ c= SS^ (^ W C^- c= SS^ (^).
(ii) jy b~(p)cz SS(u\ Q)(V7?eZ) W ;/ b ~ (p1) ^ SS (u \ Q) (V^eCi- U C^-VZ), ^^
^?+(^)c=SS(M|Q)/or^//^GCl+UC2+.
Remark. — Let C be given by (7.5) with G^QTI. Then, for any nonempty finite subset Z of Ci~ U C^", there exists peZ such that Rp 0 (C^ U C2)^Z.
7.3.3. Proof of Theorems. — To prove Theorems 7.2 and 7.3, we first prepare the following general lemma:
LEMMA 7.4. — Let u be a section of H°R ^fom^ (^x/^x P? ^Q|x) ;72 ^ neighborhood ofp e C^ 0 C^. 77^ SS^^ (^ = SS^ (^/^ a neighborhood of p .
Proof. — Let p ' eC^ 0 C^. Since ^+ (7?') is transversal to each of N^ and N3, and n(b+(p'))<^0.^ (^\Q.^ it follows from'the theorem of propagation of regularity up to smooth boundary (c/., e.g., [7], [16], [17]) that, for ^=1, 2, p'eSS^(u) if and only if b + (//) c= SS (M | Q). This completes the proof.
Since Theorem 7.2 is a special case of Theorem 7.3, it is sufficient to prove Theorem 7.3. Assertions (i) and (ii) of Theorem 7.3 are equivalent by the theorem of propagation of microanalyticity up to smooth boundary (cf. e.g., [7], [16], [17]).
Proof of Theorem 7.3. — Let Z be a finite subset of C ^ U C ^ " . Assume
c,- nss^(^)=c,- nz (^=1,2). set z^zu^ nss^^uo^ nss^)).
Step 1. — Suppose (a-1)
(a-2)
zn(cruc2-)=z,
[P^Pw]cz (V^eZ).
Then, for any p e Z, R^ c= 2 by (a-2). Hence Rp U (Ci~ U C^) c: Z n (Ci" U C^) = Z by (a-1); this is a contradiction to the assumption of the theorem. Thus we may assume
4eSERIE - TOME 25 - 1992 - N° 1
one of the following:
(b-i) zn(Ci-uc2-)^z.
(b-2) there exists p e Z such that {p^, p^} ^ Z.
Step 2. - Assume (b-1). Let peZC\ Cf\Z; then pe(C^ U SS^(u))\Z. Since peC^\Z [therefore Z?" (/?) Pi SS(^|Q)=0], it follows from Theorem 6.5 that a small neighborhood of p in C is contained in SS^ (u). Thus, applying Lemma 7.1 for u |^
on U^ = C^\Z^), we have U^ c: SS^ (u), therefore C^ (<= U^) c SS^ (^). In particular, we have C^ Fi C^ <= SS^ (u); therefore C^ 0 C^ c:SS^(^) by Lemma 7.4. Applying again Lemma 7.1 for u\^ on \J^=C^\Z^ we have C^cSS^M). As a result, C^ c SS^ (^) and C^" c= SS^ (^).
5'^ 3. — Assume (a-1). Then we may assume by (b-2) that there exists peZ such that j^^Z. Suppose peZC}C^. Then peZr\C^ by (a-1). Since p is an isolated point of ZpiCi", we have p^eC^ r}SS^(u) by Lemma 7.1(ii). Therefore p^eZ, which is a contradiction. Suppose peZf^D^. Then ^)(=^)eZ. Thus we have peZ^C^
(Case 1) Let peC^SS^(u) with ^D^Z. Since C i - n Z = C m Z by (a-1), P([)^Z. Since peC^ H SS^^ (^), by applying Lemma 7.1 (i) for u\^ on C^VZ^, we have C^ cSS^^ (^). Then, using the same argument as in Step 2 (with permutating Q^
and Q^)? we have also C^ ^SS^^ (^).
(Ca^ 2) Let ^eC^ n C^ Pi SS^(^) with ^D^Z. By Lemma 7.4, we have:
^eC^ 0 C^ 0 SS^^ (M). This is reduced to Case 1.
(Case 3) Let p e Z Pi Ci4' with p^ ^ Z. Then ;? e Z U C^ U C^; therefore p e SS^ (^).
Since p is an isolated point of C^ Pi SS^ (u), it follows from Corollary 6.3 that /?eSSn^ (u). Thus this is reduced to Case 1.
This completes the proof.
Remark. — The argument used above is not restricted to the case where C is an ellipse nor to the case of second order differential equations. One can get an analogous result for higher order differential equations by imposing many enough boundary conditions.
7.3.4. An example. — Let P be the wave operator on a C^-Riemannian manifold (M'.g) : P=Df-\, where Ag is the Laplace operator on (M', g). Let K/^cpi^O,
^P2^}? wltn ^i? ^2 being real-valued CMunctions on M' with ^(pi A^cp^O. Put M=]VTx[R^ K = K ' x l R ^ Q^M'XJK/. Syppose that u describes a simple processing curvilinear wave on Q7; i.e., u is the solution to the mixed problem
Pu==0 on Q'x ff^
^k=0.
U ^ o ^ O . Dlu\l=0=6xo^
ANNALES SCIENT1F1QU.S 1 ) 1 1 I ( 01 I \ 0 1 < M A M SCPI'.KII I K I
(p) primary wave (r) reflected wave (d) diffracted wave
Fig. 7 . 5 . — Diffraction of a simple progressing wave by a corner.
where 8^ is Dirac's 8-function on M' supported at XoeQ,'. This is one of the typical cases to which the result of 7 . 3 . 2 is applicable (cf. Fig. 7.5 for diffraction by a corner of a simple progressing curvilinear wave). Cf. Cheeger-Taylor [I], Rouleux [14], Varrenne [27]; their results (determination of the locus of singularities of the fundamental solution to the considered mixed problem) give an estimate from above to the singularities of a general solution for any initial data, but they do not seem to treat the diffraction of a single incident ray by a corner.
Remark. — It is not proved in our situation that there are no other singularities than the incident rays and the diffracted rays (cf. the results of [I], [14], [27], etc.); it is in fact possible in a certain case to construct a solution having singularities on a "diffracted cone" with no incident rays. (The author is grateful to G. Lebeau for kindly suggesting this construction.)
REFERENCES
[1] J. CHEEGER and M. TAYLOR, On the Diffraction of the Waves by Conical Singularities I - I I (Comm. P.A.M., Vol. 35, 1982, pp. 275-331 et 487-529).
[2] R. HARTSHORNE, Residues and Duality (Led. Notes Math., No. 20, Springer-Verlag, 1966).
[3] M. KASHIWARA and T. KAWAI, On the Boundary Value Problem for Elliptic System of Linear Partial Differential Equations I - I I (Proc. Jpn Acad., Vol. 48, 1972, pp. 712-715; ibid., Vol.49, 1973, pp. 164-168).
[4] M. KASHIWARA, T. KAWAI and T. KIMURA, Foundations of Algebraic Analysis, Princeton University Press, 1986.
[5] M. KASHIWARA and P. SCHAPIRA, Micro-Hyperbolic Systems (Acta Math., Vol. 142, 1979, pp. 1-55).
[6] M. KASHIWARA and P. SCHAPIRA, Microlocal Study of Sheaves (Asterisque, Vol. 128, Soc. Math. de France, 1985).
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